Optimal. Leaf size=61 \[ \frac{2}{15} b c^{5/2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{c} x\right ),-1\right )-\frac{a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}-\frac{2 b c \sqrt{1-c^2 x^4}}{15 x^3} \]
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Rubi [A] time = 0.0338484, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4842, 12, 325, 221} \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}-\frac{2 b c \sqrt{1-c^2 x^4}}{15 x^3}+\frac{2}{15} b c^{5/2} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 325
Rule 221
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^2\right )}{x^6} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac{1}{5} b \int \frac{2 c}{x^4 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac{1}{5} (2 b c) \int \frac{1}{x^4 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{2 b c \sqrt{1-c^2 x^4}}{15 x^3}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac{1}{15} \left (2 b c^3\right ) \int \frac{1}{\sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{2 b c \sqrt{1-c^2 x^4}}{15 x^3}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac{2}{15} b c^{5/2} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.133281, size = 72, normalized size = 1.18 \[ -\frac{-2 i b (-c)^{5/2} x^5 \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c} x\right ),-1\right )+3 a+2 b c x^2 \sqrt{1-c^2 x^4}+3 b \sin ^{-1}\left (c x^2\right )}{15 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 87, normalized size = 1.4 \begin{align*} -{\frac{a}{5\,{x}^{5}}}+b \left ( -{\frac{\arcsin \left ( c{x}^{2} \right ) }{5\,{x}^{5}}}+{\frac{2\,c}{5} \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{-{c}^{2}{x}^{4}+1}}+{\frac{1}{3}{c}^{{\frac{3}{2}}}\sqrt{-c{x}^{2}+1}\sqrt{c{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{c},i \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{4}+1}}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arcsin \left (c x^{2}\right ) + a}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.44533, size = 61, normalized size = 1. \begin{align*} - \frac{a}{5 x^{5}} + \frac{b c \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{c^{2} x^{4} e^{2 i \pi }} \right )}}{10 x^{3} \Gamma \left (\frac{1}{4}\right )} - \frac{b \operatorname{asin}{\left (c x^{2} \right )}}{5 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (c x^{2}\right ) + a}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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